Skip to main content
SpringerLink
Log in

Flows with unit path capacities and related packing and covering problems

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest.

Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an \(O(\sqrt{m})\) -approximation algorithm. We argue that this performance guarantee is best possible, unless P=NP.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Andrews M, Zhang L (2007) Hardness of the undirected congestion minimization problem. SIAM J Comput 37(1):112–131

    Article  MATH  MathSciNet  Google Scholar 

  • Baier G, Erlebach T, Hall A, Köhler E, Schilling H, Skutella M (2006) Length-bounded cuts and flows. In: Proceedings of the 33rd international colloquium on automata, languages and programming, pp 679–690

  • Bartholdi L (1999) Counting paths in graphs. Enseign Math 45:83–131

    MATH  MathSciNet  Google Scholar 

  • Baveja A, Srinivasan A (2000) Approximation algorithms for disjoint paths and related routing and packing problems. Math Oper Res 25:255–280

    Article  MATH  MathSciNet  Google Scholar 

  • Chekuri C, Khanna S (2007). Edge-disjoint paths revisited. ACM Trans Algorithms (TALG), 3(4)

  • Dijkstra EW (1959) A note on two problems in connection with graphs. Numer Math 1:269–271

    Article  MATH  MathSciNet  Google Scholar 

  • Dreyfus SE (1969) An appraisal of some shortest path algorithms. Oper Res 17:395–412

    Article  MATH  Google Scholar 

  • Eppstein D (1998) Finding the k shortest paths. SIAM J Comput 28:652–673

    Article  MATH  MathSciNet  Google Scholar 

  • Fleischer LK (2000) Approximating fractional multicommodity flow independent of the number of commodities. SIAM J Discrete Math 13(4):505–520

    Article  MATH  MathSciNet  Google Scholar 

  • Ford LR, Fulkerson DR (1956) Maximal flow through a network. Can J Math 8:399–404

    MATH  MathSciNet  Google Scholar 

  • Fox BL (1975) k-th shortest paths and applications to the probabilistic networks. In: ORSA/TIMS joint national meeting, 23:B263

  • Garg N, Könemann J (1998) Faster and simpler algorithms for multicommodity flow and other fractional packing problems. In: Proceedings of the 39th annual IEEE symposium on foundations of computer science, pp 300–309

  • Gessel I (1993) Counting paths in Young’s lattice. J Stat Plan Inference 34:125–134

    Article  MATH  MathSciNet  Google Scholar 

  • Grigoriadis M, Khachiyan LG (1995a) An exponential-function reduction method for block-angular convex programs. Networks 26(1.2):59–68

    Article  MATH  MathSciNet  Google Scholar 

  • Grigoriadis M, Khachiyan LG (1995b) A sublinear-time randomized approximation algorithm for matrix games. Oper Res Lett 18(2):53–58

    Article  MATH  MathSciNet  Google Scholar 

  • Grigoriadis M, Khachiyan LG (1996a) Approximate minimum-cost multicommodity flows in o(ε −2 knm) time. Math Program 75:477–482

    MathSciNet  Google Scholar 

  • Grigoriadis M, Khachiyan LG (1996b) Coordination complexity of parallel price-directive decomposition. Math Oper Res 21:321–340

    Article  MATH  MathSciNet  Google Scholar 

  • Guruswami V, Khanna S, Rajaraman R, Shepherd B, Yannakakis M (1999) Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems. In: Proceedings of the 31st annual ACM symposium on theory of computing, pp 19–28

  • Klein P, Plotkin SA, Shmoys DB, Tardos E (1994) Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts. SIAM J Comput 23(3):466–487

    Article  MATH  MathSciNet  Google Scholar 

  • Kleinberg JM (May 1996) Approximation algorithms for disjoint path problems. PhD thesis, Massachusetts Institute of Technology

  • Kolliopoulos SG (2007) Edge-disjoint paths and unsplittable flow. In: Gonzalez TF (ed) Handbook of approximation algorithms and metaheuristics. Chapman & Hall/CRC, London

    Google Scholar 

  • Kolliopoulos SG, Stein C (2004) Approximating disjoint-path problems using packing integer programs. Math Program Ser A 99:63–87

    Article  MATH  MathSciNet  Google Scholar 

  • Lawler EL (1972) A procedure for computing the K best solutions to discrete optimization problems and its application to the shortest path problem. Manag Sci 18:401–405

    Article  MATH  MathSciNet  Google Scholar 

  • Leighton T, Makedon F, Plotkin SA, Stein C, Tardos E (1995) Fast approximation algorithms for multicommodity flow problems. J Comput Syst Sci 50(2):228–243

    Article  MATH  MathSciNet  Google Scholar 

  • Lucas JF (1983) Paths and Pascal numbers. Two-Year College Math J 14(4):329–341

    Article  Google Scholar 

  • Martens M (2007) Path-constrained network flows. PhD thesis, Universität Dortmund

  • Martens M, Skutella M (2006) Flows on few paths: Algorithms and lower bounds. Networks 48(2):68–76

    Article  MATH  MathSciNet  Google Scholar 

  • Papadimitriou CH (1994) Computational complexity. Addison-Wesley, Reading

    MATH  Google Scholar 

  • Plotkin SA, Shmoys DB, Tardos E (1995) Fast approximation algorithms for fractional packing and covering problems. Math Oper Res 20:257–301

    Article  MATH  MathSciNet  Google Scholar 

  • Raghavan P, Thompson CD (1987) Randomized rounding: A technique for provably good algorithms and algorithmic proofs. Combinatorica 7:365–374

    Article  MATH  MathSciNet  Google Scholar 

  • Roditty L (2007) On the k-simple shortest paths problem in weighted directed graphs. In: Proceedings of the 18th annual ACM-SIAM symposium on discrete algorithms, pp 920–928

  • Schrijver A (2003) Combinatorial optimization. Polyhedra and efficiency. Springer, Berlin

    MATH  Google Scholar 

  • Shahrokhi F, Matula DW (1990) The maximum concurrent flow problem. J ACM 37:318–334

    Article  MATH  MathSciNet  Google Scholar 

  • Stanley RP (1996) A matrix for counting paths in acyclic digraphs. J Comb Theory Ser A 74(1):169–172

    Article  MATH  MathSciNet  Google Scholar 

  • Valiant LG (1979) The complexity of enumeration and reliability problems. SIAM J Comput 8(3):410–421

    Article  MATH  MathSciNet  Google Scholar 

  • Yen JY (1971) Finding the k shortest loopless paths in a network. Manag Sci 17(11):712–716

    Article  MATH  MathSciNet  Google Scholar 

  • Young NE (2001) Sequential and parallel algorithms for mixed packing and covering. In: IEEE symposium on foundations of computer science, pp 538–546

Download references

Author information

Authors and Affiliations

  1. Zuse Institute Berlin, Takustr. 7, 14195, Berlin, Germany

    Maren Martens

  2. TU Berlin, Institut für Mathematik, MA 5–2, Str. des 17. Juni 136, 10623, Berlin, Germany

    Martin Skutella

Authors
  1. Maren Martens
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Skutella
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maren Martens.

Additional information

This work was partially supported by the DFG Research Center Matheon in Berlin, by the Graduate School of Production Engineering and Logistics, North Rhine-Westphalia, by the DFG Focus Program 1126 and the DFG grants SK 58/4-1 and SK 58/5-3, and by an NSERC Operating Grant.

Part of this work was done while M. Martens was at Universität Dortmund and at the Sauder School of Business, University of British Columbia.

Part of this work was done while M. Skutella was at Universität Dortmund.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martens, M., Skutella, M. Flows with unit path capacities and related packing and covering problems. J Comb Optim 18, 272–293 (2009). https://doi.org/10.1007/s10878-009-9225-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-009-9225-x

Keywords

Search

Navigation